3. Strong Isomorphism in Eisert-Wilkens-Lewenstein Quantum Games
Having specified the notion of strong isomorphism and the generalized Eisert-Wilkens-Lewenstein scheme we will now check if the isomorphism between the classically played games makes the corresponding quantum games isomorphic. We first examine the case when the players’ unitary strategies depend on two parameters. The quantum game
Γ
EWL
with
(14)
D
i
=
U
i
θ
i
,
α
i
,
0
:
θ
i
∈
0
,
π
,
α
i
∈
0,2
π
is particularly interested. That setting was used to introduce the EWL scheme [2] and has been widely studied in recent years (see, e.g., [7–10]). We begin with an example of isomorphic games that describe the Prisoner’s Dilemma.
Example 8.
The generalized Prisoner’s Dilemma game and one of its isomorphic counterparts may be given by the following bimatrices:
(15)
Γ
:
t
b
l
R
,
R
T
,
S
r
S
,
T
P
,
P
,
Γ
′
:
t
′
b
′
l
′
S
,
T
P
,
P
r
′
R
,
R
T
,
S
,
where
T
>
R
>
P
>
S
. Note that the games are the same up to the order of player 2’s strategies. Let us now examine the EWL approach to
Γ
and
Γ
′
defined by triples
(16)
Γ
EWL
=
1,2
,
U
i
θ
i
,
α
i
,
0
i
∈
1,2
,
M
i
i
∈
1,2
,
Γ
EWL
′
=
1,2
,
U
i
′
θ
i
′
,
α
i
′
,
0
i
∈
1,2
,
M
i
′
i
∈
1,2
,
where
(17)
M
1
,
M
2
=
R
,
R
00
00
+
S
,
T
01
01
+
T
,
S
10
10
+
P
,
P
11
11
,
M
1
′
,
M
2
′
=
S
,
T
00
00
+
R
,
R
01
01
+
P
,
P
10
10
+
T
,
S
11
11
.
We first compare the sets of Nash equilibria in
Γ
EWL
and
Γ
EWL
′
to check if the games may be isomorphic. We recall from [2] that there is the unique Nash equilibrium
U
1
(
0
,
π
/
2,0
)
⊗
U
2
(
0
,
π
/
2,0
)
in
Γ
EWL
that determines the payoff profile
(
R
,
R
)
. When it comes to
Γ
EWL
′
, we set
n
=
2
in (11) and replace (12) by
M
1
′
and
M
2
′
from (17). Then we can rewrite (13) as
(18)
u
1
′
,
u
2
′
U
1
θ
1
′
,
α
1
′
,
0
⊗
U
2
θ
2
′
,
α
2
′
,
0
=
S
,
T
c
o
s
α
1
′
+
α
2
′
cos
θ
1
′
2
cos
θ
2
′
2
2
+
R
,
R
cos
α
1
′
cos
θ
1
′
2
sin
θ
2
′
2
+
sin
α
2
′
sin
θ
1
′
2
cos
θ
2
′
2
2
+
P
,
P
sin
α
1
′
cos
θ
1
′
2
sin
θ
2
′
2
+
cos
α
2
′
sin
θ
1
′
2
cos
θ
2
′
2
2
+
T
,
S
sin
α
1
′
+
α
2
′
cos
θ
1
′
2
cos
θ
2
′
2
-
sin
θ
1
′
2
sin
θ
2
′
2
2
.
Let
U
2
′
(
θ
2
′
,
α
2
′
,
0
)
be an arbitrary but fixed strategy of player 2. Then it follows from (18) that strategy
U
1
′
(
θ
1
′
,
α
1
′
,
0
)
specified by equation
(19)
U
1
′
θ
1
′
,
α
1
′
,
0
=
U
1
′
θ
2
′
,
3
π
2
-
α
2
′
,
0
if
α
2
′
∈
0
,
3
π
2
,
U
1
′
θ
2
′
,
7
π
2
-
α
2
′
,
0
if
α
2
′
∈
3
π
2
,
2
π
is player 1’s best reply to
U
2
′
(
θ
2
′
,
α
2
′
,
0
)
as it yields player 1’s payoff
T
. Hence, a possible Nash equilibrium would generate the maximal payoff for player 1. On the other hand, given a fixed player 1’s strategy
U
1
′
(
θ
1
′
,
α
1
′
,
0
)
, player 2 can obtain a payoff that is strictly higher than
S
by choosing, for example,
U
2
(
θ
2
′
,
α
2
′
,
0
)
with
θ
2
′
=
0
,
α
2
′
=
2
π
-
α
1
′
. This means that player 1 would obtain strictly less than
T
. Hence, there is no pure Nash equilibrium in the game determined by
Γ
EWL
′
. As a result, we can conclude by Lemma 7 that games (16) are not strongly isomorphic.
The example given above shows that the EWL approach with the two-parameter unitary strategies may output different Nash equilibria depending on the order of players’ strategies in the classical game. This appears to be a strange feature since games (15) represent the same decision problem from a game-theoretical point of view.
One way to make games (16) isomorphic is to replace player
i
’s strategy set (14) with the alternative two-parameter strategy space
(20)
F
i
=
U
i
θ
i
,
0
,
β
i
:
θ
i
∈
0
,
π
,
β
i
∈
0,2
π
every time player
i
’s strategies are switched in the classical game. In the case of games (16) this means that quantum games
(21)
Γ
EWL
=
1,2
,
D
1
,
D
2
,
M
i
i
∈
1,2
,
Γ
EWL
′
=
1,2
,
D
1
′
,
F
2
′
,
M
i
′
i
∈
1,2
are isomorphic. Indeed, define a game map
f
~
=
(
η
,
φ
~
1
,
φ
~
2
)
with
η
(
i
)
=
i
for
i
=
1,2
and bijections
φ
~
1
:
D
1
→
D
1
′
and
φ
~
2
:
D
2
→
F
2
′
satisfying
(22)
φ
~
1
U
1
θ
1
,
α
1
,
0
=
U
1
′
θ
1
,
α
1
,
0
,
φ
~
2
U
2
θ
1
,
α
1
,
0
=
U
2
′
π
-
θ
2
,
0
,
π
-
α
2
.
The map
φ
~
2
should actually distinguish cases
α
2
∈
[
0
,
π
)
and
α
2
∈
[
π
,
2
π
)
to be a well-defined bijection as it was done in (19). To simplify the proof we stick to the form (22) throughout the paper bearing in mind that for
π
-
α
2
∉
[
0,2
π
)
we can always find the equivalent angle
3
π
-
α
2
∈
[
0,2
π
)
. We have to show for games (21) that
(23)
u
i
U
1
⊗
U
2
=
Ψ
M
i
Ψ
=
Ψ
′
M
i
′
Ψ
′
=
u
i
′
f
~
U
1
⊗
U
2
for
i
=
1,2
, where
|
Ψ
〉
=
J
†
(
U
1
⊗
U
2
)
J
|
00
〉
and
Ψ
′
=
J
†
(
f
~
(
U
1
⊗
U
2
)
)
J
|
00
〉
. First, note that
U
2
′
(
π
-
θ
2
,
0
,
π
-
α
2
)
=
i
σ
x
U
2
′
(
θ
2
,
α
2
,
0
)
. Hence, we obtain
(24)
Ψ
′
=
J
†
f
~
U
1
θ
1
,
α
1
,
0
⊗
U
2
θ
2
,
α
2
,
0
J
00
=
J
†
U
1
′
θ
1
,
α
1
,
0
⊗
U
2
π
-
θ
2
,
0
,
π
-
α
2
J
00
=
1
⊗
-
i
σ
x
J
†
U
1
′
θ
1
,
α
1
,
0
⊗
U
2
′
θ
2
,
α
2
,
0
J
00
=
1
⊗
-
i
σ
x
Ψ
.
Application of (24) finally yields
(25)
Ψ
′
M
i
′
Ψ
′
=
Ψ
1
⊗
i
σ
x
M
i
′
1
⊗
-
i
σ
x
Ψ
=
Ψ
M
1
Ψ
.
In similar way we can prove a more general fact. Namely, if
F
2
is player 2’s strategy set in one of games (16) and
D
2
is in the the other one then games (16) become strongly isomorphic. This observation suggests that the EWL scheme is robust with respect to changing the order of players’ strategies in the classical game if the players can use strategies from
D
i
∪
F
i
or equivalently from the set
S
U
(
2
)
. Before stating the general result we study a specific example.
Example 9.
Let us consider the following three-person games:
(26)
v
t
b
l
a
000
,
b
000
,
c
000
a
100
,
b
100
,
c
100
r
a
010
,
b
010
,
c
010
a
110
,
b
110
,
c
110
,
w
t
b
l
a
001
,
b
001
,
c
001
a
101
,
b
101
,
c
101
r
a
011
,
b
011
,
c
011
a
111
,
b
111
,
c
111
,
v
′
t
′
b
′
l
′
c
011
,
a
011
,
b
011
c
010
,
a
010
,
b
010
r
′
c
111
,
a
111
,
b
111
c
110
,
a
110
,
b
110
,
w
′
t
′
b
′
l
′
c
001
,
a
001
,
b
001
c
000
,
a
000
,
b
000
r
′
c
101
,
a
101
,
b
101
c
100
,
a
100
,
b
100
.
The games are (strongly) isomorphic via game mapping
f
=
(
η
,
φ
1
,
φ
2
,
φ
3
)
such that
(27)
η
=
1
⟶
2,2
⟶
3,3
⟶
1
,
φ
1
=
t
⟶
l
′
,
b
⟶
r
′
,
φ
2
=
l
⟶
w
′
,
r
⟶
v
′
,
φ
3
=
v
⟶
b
′
,
w
⟶
t
′
.
We see from (27) that the isomorphism maps strategy profiles as follows:
(28)
f
t
,
l
,
v
=
b
′
,
l
′
,
w
′
,
f
t
,
r
,
v
=
b
′
,
l
′
,
v
′
,
f
b
,
l
,
v
=
b
′
,
r
′
,
w
′
,
f
b
,
r
,
v
=
b
′
,
r
′
,
v
′
,
f
t
,
l
,
w
=
t
′
,
l
′
,
w
′
,
f
t
,
r
,
w
=
t
′
,
l
′
,
v
′
,
f
b
,
l
,
w
=
t
′
,
r
′
,
w
′
,
f
b
,
r
,
w
=
t
′
,
r
′
,
v
′
.
Let us now define the EWL quantum extensions
Γ
EWL
and
Γ
EWL
′
for the three-player game where we identify the players’ first and second strategies with values 0 and 1, respectively. That is,
(29)
Γ
EWL
=
N
,
D
i
i
∈
N
,
M
i
i
∈
N
,
Γ
EWL
′
=
N
,
D
i
′
i
∈
N
,
M
i
′
i
∈
N
,
where
N
=
{
1,2
,
3
}
,
D
i
=
D
i
′
=
S
U
(
2
)
for each
i
∈
N
,
(30)
M
1
,
M
2
,
M
3
=
∑
j
1
,
j
2
,
j
3
=
0,1
a
j
1
j
2
j
3
,
b
j
1
j
2
j
3
,
c
j
1
j
2
j
3
P
j
1
j
2
j
3
,
M
1
′
,
M
2
′
,
M
3
′
=
∑
j
1
,
j
2
,
j
3
=
0,1
c
j
1
j
2
j
3
,
a
j
1
j
2
j
3
,
b
j
1
j
2
j
3
P
f
j
1
j
2
j
3
,
where
P
j
1
j
2
j
3
=
j
1
j
2
j
3
j
1
j
2
j
3
. Given
f
=
(
η
,
(
φ
i
)
i
∈
N
)
let us define a mapping
f
~
=
(
η
,
(
φ
~
i
)
i
∈
N
)
such that
φ
~
i
:
D
i
→
D
η
i
′
for
i
∈
N
and
(31)
φ
~
1
U
1
θ
1
,
α
1
,
β
1
=
U
2
′
θ
1
,
α
1
,
β
1
,
φ
~
2
U
2
θ
2
,
α
2
,
β
2
=
U
3
′
π
-
θ
2
,
2
π
-
β
2
,
π
-
α
2
,
φ
~
3
U
3
θ
3
,
α
3
,
β
3
=
U
1
′
π
-
θ
3
,
2
π
-
β
3
,
π
-
α
3
.
Then,
f
~
induces a bijection from
D
1
⊗
D
2
⊗
D
3
to
D
1
′
⊗
D
2
′
⊗
D
3
′
such that
(32)
f
~
U
1
⊗
U
2
⊗
U
3
=
φ
~
3
U
3
θ
3
,
α
3
,
β
3
,
φ
~
1
U
1
θ
1
,
α
1
,
β
1
⊗
φ
~
2
U
2
θ
2
,
α
2
,
β
2
=
U
1
′
π
-
θ
3
,
2
π
-
β
3
,
π
-
α
3
⊗
U
2
′
θ
1
,
α
1
,
β
1
⊗
U
3
′
π
-
θ
2
,
2
π
-
β
2
,
π
-
α
2
.
According to the EWL scheme, the payoff functions for
Γ
EWL
and
Γ
EWL
′
are as follows:
(33)
u
i
U
1
⊗
U
2
⊗
U
3
=
Ψ
M
i
Ψ
,
w
h
e
r
e
Ψ
=
J
†
U
1
⊗
U
2
⊗
U
3
J
000
,
u
i
′
U
1
′
⊗
U
2
′
⊗
U
3
′
=
Ψ
′
M
i
′
Ψ
′
,
w
h
e
r
e
Ψ
′
=
J
†
U
1
′
⊗
U
2
′
⊗
U
3
′
J
000
for
i
∈
N
. In order to prove that
Γ
EWL
and
Γ
EWL
′
are isomorphic we have to check if
(34)
u
i
U
1
⊗
U
2
⊗
U
3
=
u
η
i
′
f
~
U
1
⊗
U
2
⊗
U
3
for
i
∈
N
.
Without loss of generality we can assume that
i
=
1
. Let us first evaluate state
Ψ
′
,
(35)
Ψ
′
=
J
†
U
1
′
π
-
θ
3
,
2
π
-
β
3
,
π
-
α
3
⊗
U
2
′
θ
1
,
α
1
,
β
1
⊗
U
3
′
π
-
θ
2
,
2
π
-
β
2
,
π
-
α
2
J
000
.
Note that
(36)
U
1
′
π
-
θ
3
,
2
π
-
β
3
,
π
-
α
3
⊗
U
2
′
θ
1
,
α
1
,
β
1
⊗
U
3
′
π
-
θ
2
,
2
π
-
β
2
,
π
-
α
2
=
-
σ
x
⊗
1
⊗
σ
x
U
1
′
θ
3
,
α
3
,
β
3
⊗
U
2
′
θ
1
,
α
1
,
β
1
⊗
U
3
′
θ
2
,
β
2
,
α
2
,
U
1
′
θ
3
,
α
3
,
β
3
⊗
U
2
′
θ
1
,
α
1
,
β
1
⊗
U
3
′
θ
2
,
α
2
,
β
2
=
S
η
U
2
′
θ
1
,
α
1
,
β
1
⊗
U
3
′
θ
2
,
α
2
,
β
2
⊗
U
1
′
θ
3
,
α
3
,
β
3
S
η
†
,
where
S
η
is a permutation matrix that changes the order of qubits according to
η
,
(37)
S
η
=
000
000
+
001
010
+
010
100
+
011
110
+
100
001
+
101
011
+
110
101
+
111
111
.
Using (36), the fact that
(38)
J
†
,
-
σ
x
⊗
1
⊗
σ
x
=
J
†
,
S
η
=
J
,
S
η
=
0
and
S
η
†
|
000
〉
=
|
000
〉
we may write
Ψ
′
as follows:
(39)
Ψ
′
=
-
σ
x
⊗
1
⊗
σ
x
S
η
J
†
U
2
′
θ
1
,
α
1
,
β
1
⊗
U
3
′
θ
2
,
α
2
,
β
2
⊗
U
1
′
θ
3
,
α
3
,
β
3
J
000
=
-
σ
x
⊗
1
⊗
σ
x
S
η
Ψ
.
Note that
j
1
j
2
j
3
S
η
=
S
η
†
j
1
j
2
j
3
†
. This means that
S
η
is the inverse operation when acting on dual vectors. This observation together with the fact that
f
changes the strategy order for players 1 and 3 leads us to conclusion that operator
±
(
σ
x
⊗
1
⊗
σ
x
)
S
η
can be viewed as
f
-
1
in the sense of the following equality:
(40)
j
1
j
2
j
3
-
σ
x
⊗
1
⊗
σ
x
S
η
=
f
-
1
j
1
j
2
j
3
.
Let us now consider term
Ψ
′
P
f
(
j
1
j
2
j
3
)
Ψ
′
for
Ψ
′
given by (35). From (39) and (40) it follows that
(41)
Ψ
′
P
f
j
1
j
2
j
3
Ψ
′
=
f
j
1
j
2
j
3
Ψ
′
2
=
f
j
1
j
2
j
3
σ
x
⊗
1
⊗
σ
x
S
η
Ψ
2
=
j
1
j
2
j
3
Ψ
2
=
Ψ
P
j
1
j
2
j
3
Ψ
.
Hence,
(42)
u
η
1
f
~
U
1
⊗
U
2
⊗
U
3
=
Ψ
′
M
η
1
′
Ψ
′
=
Ψ
M
1
Ψ
=
u
1
U
1
⊗
U
2
⊗
U
3
.
Similar reasoning applies to the case
i
=
2,3
. We have thus proved that games given by (29) are isomorphic.
The same conclusion can be drawn for games with arbitrary but finite number
N
of players.
Proposition 10.
Let
Γ
=
(
N
,
(
S
i
)
i
∈
N
,
(
u
i
)
i
∈
N
)
and
Γ
′
=
(
N
,
(
S
i
′
)
i
∈
N
,
(
u
i
′
)
i
∈
N
)
be strongly isomorphic strategic form games with
|
S
i
|
=
|
S
i
′
|
=
2
and let
Γ
EWL
=
(
N
,
(
D
i
)
i
∈
N
,
(
M
i
)
i
∈
N
)
and
Γ
EWL
′
=
(
N
,
(
D
i
′
)
i
∈
N
,
(
M
i
′
)
i
∈
N
)
with
D
i
=
D
i
′
=
S
U
(
2
)
be the corresponding quantum games. Then
Γ
EWL
and
Γ
EWL
′
are strongly isomorphic.
Proof.
The proof follows by the same method as in Example 9. Let
f
=
(
η
,
(
φ
i
)
i
∈
N
)
be a strong isomorphism between
Γ
and
Γ
′
. Depending on
φ
i
:
S
i
→
S
η
i
′
such that
φ
i
(
s
k
i
)
=
s
l
η
(
i
)
for
A
i
=
{
s
0
i
,
s
1
i
}
and
A
η
(
i
)
=
{
s
0
η
(
i
)
,
s
1
η
(
i
)
}
we construct
f
~
=
(
η
,
(
φ
~
i
)
i
∈
N
)
, where
(43)
φ
~
i
U
i
θ
i
,
α
i
,
β
i
=
U
η
i
′
θ
i
,
α
i
,
β
i
if
φ
i
s
k
i
=
s
k
η
i
U
η
i
′
π
-
θ
i
,
2
π
-
β
i
,
π
-
α
i
if
φ
i
s
k
i
=
s
k
⊕
2
1
η
i
.
Then
f
~
⨂
i
=
1
N
U
i
=
⨂
i
=
1
N
U
i
′
, where
U
η
i
′
=
φ
~
i
(
U
i
)
for
i
=
1
,
…
,
N
. Since
η
is a permutation and
U
(
π
-
θ
,
2
π
-
β
,
π
-
α
)
=
-
i
σ
x
U
(
θ
,
α
,
β
)
, we can write relation (43) as
(44)
φ
~
η
-
1
i
U
η
-
1
i
θ
η
-
1
i
,
α
η
-
1
i
,
β
η
-
1
i
=
U
i
′
θ
η
-
1
i
,
α
η
-
1
i
,
β
η
-
1
i
if
φ
i
s
k
i
=
s
k
η
i
-
i
σ
x
U
i
′
θ
η
-
1
i
,
α
η
-
1
i
,
β
η
-
1
i
if
φ
i
s
k
i
=
s
k
⊕
2
1
η
i
.
As a result,
f
~
maps
⨂
i
=
1
N
U
η
-
1
(
i
)
onto
⨂
i
=
1
N
U
i
′
as follows:
(45)
f
~
⨂
i
=
1
N
U
i
=
⨂
i
=
1
N
V
i
⨂
i
=
1
N
U
i
′
θ
η
-
1
i
,
α
η
-
1
i
,
β
η
-
1
i
,
V
i
=
1
if
φ
i
s
k
i
=
s
k
η
i
-
i
σ
x
if
φ
i
s
k
i
=
s
k
⊕
2
1
η
i
.
Let us now consider a permutation matrix
S
η
∈
M
2
N
that rearranges the order of basis states
{
|
j
i
〉
}
∈
{
|
0
〉
,
|
1
〉
}
in the tensor product
j
1
j
2
,
…
,
|
j
N
〉
. Since
S
η
permutes the elements in a similar way as
f
~
, it is not difficult to see that
(46)
S
η
⨂
i
=
1
N
U
i
θ
i
,
α
i
,
β
i
S
η
T
=
⨂
i
=
1
N
U
i
θ
η
-
1
i
,
α
η
-
1
i
,
β
η
-
1
i
.
It is also clear that
σ
x
⊗
N
commutes with
⨂
i
=
1
N
V
i
and
S
η
and so does
J
=
(
1
⊗
N
+
i
σ
x
⊗
N
)
/
2
. Thus the final state
Ψ
′
=
J
†
f
~
⨂
i
=
1
N
U
i
(
θ
i
,
α
i
,
β
i
)
J
0
⊗
N
may be written as
(47)
Ψ
′
=
⨂
i
=
1
N
V
i
S
η
J
†
⨂
i
=
1
N
U
i
θ
i
,
α
i
,
β
i
J
0
⊗
N
=
⨂
i
=
1
N
V
i
S
η
Ψ
.
Analysis similar to that in (40)–(42) shows that
(48)
u
η
i
f
~
⨂
i
=
1
N
U
i
=
u
i
⨂
i
=
1
N
U
i
,
which is the desired conclusion.
As the following example shows, the converse is not true in general.
Example 11.
Let us consider two 2 × 2 bimatrix games that differ only in the order of payoff profiles in the antidiagonal; that is,
(49)
Γ
:
t
b
l
a
00
,
b
00
a
10
,
b
10
r
a
01
,
b
01
a
11
,
b
11
,
Γ
′
:
t
′
b
′
l
′
a
00
,
b
00
a
01
,
b
01
r
′
a
10
,
b
10
a
11
,
b
11
.
The EWL quantum counterparts
Γ
E
W
L
and
Γ
E
W
L
′
for these games are specified by triples (29), where in this case
N
=
{
1,2
}
,
D
i
=
D
i
′
=
S
U
(
2
)
, and the measurement operators take the form
(50)
M
1
,
M
2
=
∑
j
1
,
j
2
=
0,1
a
j
1
j
2
,
b
j
1
j
2
P
j
1
j
2
,
M
1
′
,
M
2
′
=
∑
j
1
,
j
2
=
0,1
a
j
1
j
2
,
b
j
1
j
2
P
j
2
j
1
,
where
P
j
1
j
2
=
j
1
j
2
j
1
j
2
. Let us set a mapping
f
~
=
(
η
,
(
φ
~
1
,
φ
~
2
)
)
with
η
(
i
)
=
i
and
φ
~
i
(
U
i
(
θ
i
,
α
i
,
β
i
)
=
U
i
′
(
π
-
θ
i
,
π
/
4
-
β
i
,
π
/
4
-
α
i
)
) for
i
=
1,2
. An easy computation shows that
(51)
Ψ
′
=
J
†
f
~
U
1
⊗
U
2
J
00
=
J
†
U
1
π
-
θ
1
,
π
4
-
β
1
,
π
4
-
α
1
⊗
U
2
π
-
θ
2
,
π
4
-
β
2
,
π
4
-
α
2
J
00
=
S
F
J
†
U
1
θ
1
,
α
1
,
β
1
⊗
U
2
θ
2
,
α
2
,
β
2
J
00
=
S
F
Ψ
,
where
S
has the outer product representation
S
=
00
00
+
01
10
+
10
01
+
11
〈
11
|
and
F
=
00
00
+
01
10
+
10
01
-
11
11
. Application of (51) gives
(52)
u
i
′
f
~
U
1
⊗
U
2
=
Ψ
′
M
i
′
Ψ
′
=
Ψ
F
S
M
i
′
S
F
Ψ
=
Ψ
M
i
Ψ
=
u
i
U
1
⊗
U
2
.
As a result, games produced by
Γ
E
W
L
and
Γ
E
W
L
′
are strongly isomorphic. This fact, however, is not sufficient to guarantee the isomorphism between
Γ
and
Γ
′
. Indeed, one can check that there is no
f
=
(
η
,
(
φ
1
,
φ
2
)
)
to satisfy
u
i
(
s
)
=
u
η
i
′
(
f
(
s
)
)
for each
s
∈
{
t
,
b
}
×
{
l
,
r
}
and
i
=
1,2
. Alternatively, given specific payoff profiles
(
a
00
,
b
00
)
=
(
4,4
)
,
(
a
01
,
b
01
)
=
(
1,3
)
,
(
a
10
,
b
10
)
=
(
3,1
)
,
(
a
11
,
b
11
)
=
(
2,2
)
, we can find three Nash equilibria in the game
Γ
and just one in the game
Γ
′
. Hence, by Lemma 7 games (49) are not isomorphic.